CBSE Class 10th Introduction To Trigonometry Details & Preparations Downloads

In the vast landscape of mathematics, there exists a powerful branch that holds the key to understanding the intricate relationships between angles and sides. Welcome to the world of trigonometry, where triangles become the building blocks for unraveling the mysteries of the universe. Trigonometry, derived from the Greek words "trigonon" (triangle) and "metron" (measure), serves as a mathematical toolkit essential for countless scientific and engineering applications.

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Trigonometric Ratios

Opposite & Adjacent Sides in a Right-Angled Triangle

In the ΔABC right-angled at B, BC is the side opposite to ∠A, AC is the hypotenuse, and AB is the side adjacent to ∠A.

Trigonometric Ratios

For the right ΔABC, right-angled at ∠B, the trigonometric ratios of the ∠A are as follows:

• sin A=opposite side/hypotenuse=BC/AC
• cos A=adjacent side/hypotenuse=AB/AC
• tan A=opposite side/adjacent side=BC/AB
• cosec A=hypotenuse/opposite side=AC/BC
• sec A=hypotenuse/adjacent side=AC/AB
• cot A=adjacent side/opposite side=AB/BC

Relation between Trigonometric Ratios

• cosec θ =1/sin θ

• sec θ = 1/cos θ

• tan θ = sin θ/cos θ

• cot θ = cos θ/sin θ=1/tan θ

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Visualization of Trigonometric Ratios Using a Unit Circle

Draw a circle of the unit radius with the origin as the center. Consider a line segment OP joining a point P on the circle to the center, which makes an angle θ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

• sin θ=PQ/OP=PQ/1=PQ

• cos θ=OQ/OP=OQ/1=OQ

• tan θ=PQ/OQ=sin θ/cos θ

• cosec θ=OP/PQ=1/PQ

• sec θ=OP/OQ=1/OQ

• cot θ=OQ/PQ=cos θ/sin θ

Trigonometric Ratios of Specific Angles

The specific angles that are defined for trigonometric ratios are 0°, 30°, 45°, 60° and 90°.

Trigonometric Ratios of 45°

If one of the angles of a right-angled triangle is 45°, then another angle will also be equal to 45°.

Let us say ABC is a right-angled triangle at B, such that;

∠ A = ∠ C = 45°

Thus, BC = AB = a (say)

Using Pythagoras theorem, we have;

AC² = AB² + BC²

= a² + a²

= 2a²

AC = a√2

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Now, from the trigonometric ratios, we have

• sin 45° = (Opp. side to angle 45°)/Hypotenuse = BC/AC = a/a√2 = 1/√2
• cos 45° = (Adj. side to angle 45°)/Hypotenuse = AB/AC = a/a√2 = 1/√2
• tan 45° = BC/AB = a/a = 1

Similarly

• cosec 45° = 1/sin 45° = √2
• sec 45° = 1/cos 45° = √2
• cot 45° = 1/tan 45° = 1

The Historical Tapestry

Trigonometry's roots stretch far back in time, intertwining with the great civilizations of antiquity. Ancient astronomers, such as the Greeks and Indians, were among the first to recognize the significance of angles in the cosmos. From measuring the heights of towering pyramids to predicting celestial events, trigonometry became an indispensable tool for understanding the physical world.

The Language of Triangles

At its core, trigonometry revolves around the study of triangles. Whether you're dealing with right-angled triangles or more complex configurations, trigonometric functions like sine, cosine, and tangent emerge as the fundamental building blocks. These functions form the backbone of trigonometry, allowing us to relate the angles of a triangle to the lengths of its sides.

Range of Trigonometric Ratios from 0 to 90 Degrees

For 0∘≤θ≤90∘,

• 0 ≤ sin θ ≤ 1

• 0 ≤ cos θ ≤ 1

• 0 ≤ tan θ < ∞
  1 ≤ sec θ < ∞

• 0 ≤ cot θ < ∞

• 1 ≤ cosec θ < ∞

tan θ and sec θ are not defined at  90∘.

cot θ and cosec θ are not defined at 0∘.

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FAQ

Q1. What is Trigonometry?

Ans Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It explores the properties of these geometric elements and provides a set of functions to analyze and solve problems involving angles and distances.

Q2. Why is Trigonometry Important?

Ans Trigonometry is essential in various fields, such as physics, engineering, architecture, and astronomy. It helps in measuring distances, analyzing forces, designing structures, and understanding the behavior of waves and oscillations.

Q3. What are the Basic Trigonometric Functions?

Ans The fundamental trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a triangle to the lengths of its sides and are crucial in solving trigonometric equations and applications.

Q4. How is Trigonometry Used in Real Life?

Ans Trigonometry has numerous real-world applications. It is used in navigation for calculating distances and directions, in physics to analyze waves and oscillations, and in architecture and engineering for designing structures and determining forces.

Q5. Can Trigonometry be Fun to Learn?

Ans Absolutely! Trigonometry might seem challenging at first, but many find it enjoyable once they grasp its concepts. Practical applications and hands-on problem-solving can make learning trigonometry a rewarding and engaging experience.